Integration Of Trigonometry Functions
Introduction for trigonometry integration:
Trigonometry is a branch of the mathematics that studies triangles, particularly right triangles. Trigonometry deals with the relationships between the sides and the angles of triangles, and with trigonometric functions, which describe those relationships and angles in general, and the motion of waves such as sound and light waves.
Integration is an important concept in the mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function of the real variable x and an interval [a, b] of the real line, the definite integration'int'f(x)dx (Source.Wikipedia)
Some Integration of Trigonometry Functions:
'int' sinx dx = cosx + c
'int'cos xdx =sinx +c
'int'cosec2x dx = cotx + c
'int'sec2x dx = tanx + c
'int'secx tanx dx = sec x + c
'int'cosecx cotx dx = cosecx + c
'int'sin(ax + b)dx = 1/a cos (ax + b) + c
'int'cos (ax + b) dx =1/a sin (ax + b) + c
'int'sec2(ax + b)dx =1/a tan (ax + b) + c
'int'cosec2 (ax + b) dx = 1/a cot (ax + b) + c
'int'cosec (ax + b) cot (ax + b) dx = 1/a cosec (ax + b) + c
Examples for Trigonometry Integration:
Example 1:
Integrate:(1)sinxcos2x (2)cotxsinx (3)1/sin2x
Solution:
(1)'int' sinx/cos2xdx = 'int' (sinx /cosx).(1/cosx )dx = 'int' tanx secx dx = secx + c
(2) 'int' cotx/sinx dx = 'int' cosecx cotx dx = cosecx + c
(3)1/sin2xdx ='int' cosec2x dx = cotx + c
Example 2:
Integrate the following with respect to x. (1) sin (lx + m) (2) sec2 (p qx) (3) cosec (4x + 3) cot (4x + 3)
Solution:
'int' sin (lx + m) dx = 1l [ cos (lx + m)] + c= 1/l cos (lx + m) + c
(2)'int' sec2 (p qx) dx = 1/q [tan(p qx)] + c
(3) 'int' cosec (4x + 3) cot (4x + 3) dx = 1/4 cosec (4x + 3) + c
Example3:
integrate sin5x cos 2x dx
Solution:
sin5x cos 2x dx = 'int' 1/2 [sin (5x + 2x) + sin (5x 2x)] dx [ 2sin A cosB = sin (A+B) + sin(A B)]
=1/2 'int' [sin7x + sin3x] dx
=1/2 [( cos7x )/ 7- ( cos3x) / 3] + c
'int' sin5x cos 2x dx = 1/2[( cos7x) / 7 + (cos3x )/ 3 ]+ c
Practice Problem for Trigonometry Integration:
integrate 'int' (tanx + cotx) 2dx
Answer: tanx cotx + c
2. 'int' cos3x dx
Answer:1/4 [3 sin x +sin 3x/3 ]+ c
by: jeri
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